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Planetary Dynamics
§ 13.4–13.8
Closed Orbits
• Ug + Ktr = constant < 0
• The closer the satellite is to the main body, the
faster it moves
• Objects do not meet at rest without a nonconservative force acting (crash!)
0
E
r
E
Ug
Escape Speed
• if an object’s E = Ug + Ktr ≥ 0, it can
“escape” from a gravitational field.
• The force never disappears, but the object
does not return.
E
0
E
r
Ug
Example Problem
Calculate the escape speed of Earth, that is,
the speed with which a rocket must be
launched if it is to completely escape the
gravitational pull of Earth. (This means that
the kinetic energy is zero when it reaches a
distance of infinity.)
Mearth = 5.976  1024 kg
Rearth = 6378 km
G = 6.67  10–11 Nm2/kg2
Group Question
Would it be possible for a planet to have the
same surface gravitational field as earth, but
a different escape speed from the surface?
Orbits
• E > 0: Open
– hyperbola
• E = 0: Escape speed
– parabola
• E < 0: Closed
– ellipse
Group Work
Find the tangential speed v of an object in
circular orbit a distance r from an attractor of
mass M.
– Use the fact that the centripetal force is the
force of gravity.
Find the square of the orbital period T2 as
well.
– Use the fact that v = 2pr/T.
Kepler’s Laws
of planetary motion
1. Planets travel in elliptical paths with one
focus at the Sun.
2. At all times, a planet’s path traces out
equal areas.
3. The square of a planet’s orbital period is
directly proportional to the cube of the
semi-major axis of the orbit.
Kepler’s Laws
2. Equal-area law
3. T2  a3
– For planets, y and AU are easy units
Kepler’s laws
• Work for other systems too (planets and
moons, etc.)
Example Problem
Calculate the orbital distance of the Moon
based on the observation that the Moon
orbits the Earth once every 27.3 days. The
mass of the Moon is 7.35  1022 kg and the
mass of the Earth is 6.01024 kg.
Double Systems
• Both objects orbit system’s center of mass
• Orbital radii < separation
• Centripetal force = gravity
Weightlessness
• Actually free fall
• No force opposes gravity
Example Problem
The planet Uranus has a radius of 25,362 km and
a surface gravity of 8.87 N/kg at its poles. Its
moon Miranda is in a circular orbit at a distance of
129,560 km from Uranus’s center. Miranda has a
mass of 6.6  1019 kg and a radius of 235 km.
a. Calculate the mass of Uranus from the given data.
Example Problem
The planet Uranus has a radius of 25,362 km and
a surface gravity of 8.87 N/kg at its poles. Its
moon Miranda is in a circular orbit at a distance of
129,560 km from Uranus’s center. Miranda has a
mass of 6.6  1019 kg and a radius of 235 km.
b. Calculate the magnitude of Miranda’s orbital
acceleration due to its orbital motion about Uranus.
Example Problem
The planet Uranus has a radius of 25,362 km and
a surface gravity of 8.87 N/kg at its poles. Its
moon Miranda is in a circular orbit at a distance of
129,560 km from Uranus’s center. Miranda has a
mass of 6.6  1019 kg and a radius of 235 km.
c. Calculate the acceleration due to Miranda’s gravity
at the surface of Miranda.
Example Problem
The planet Uranus has a radius of 25,362 km and
a surface gravity of 8.87 N/kg at its poles. Its
moon Miranda is in a circular orbit at a distance of
129,560 km from Uranus’s center. Miranda has a
mass of 6.6  1019 kg and a radius of 235 km.
d. Do the answers to parts b and c mean that an object
released 1 m above Miranda’s surface on the side
toward Uranus will fall up relative to Miranda?
Explain what is happening.